Are these lattices distributive?

Is the poset of positive integers ordered by divisibility a distributive lattice? If so, can you interpret it as the lattice of downsets of a poset? Which poset? (Strictly speaking, Birkhoff’s Theorem doesn’t apply to infinite lattices.)

Is the poset of real numbers with the standard order a distributive lattice? If so, can you interpret it as the lattice of downsets of a poset? Which poset?

 
Is the poset of subgroups of an arbitrary group ordered by containment a distributive lattice? If so, can you interpret it as the lattice of downsets of a poset? Which poset?

 

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4 comments

  1. The poset of positive integers ordered by divisibility is a distributive lattice. You can treat it as a lattice of subsets ordered by containment, where you’ve treated repeated identical elements as being unique – the factors of a number would then be the elements of a subset. Then distributivity is clear by the distributivity of union and intersection, as per the fundamental laws of set algebra. Then each integer in the above poset can represent a downset of the power set of a given set (that is, the set of factors of a number as described above), ordered by containment.

  2. Oh, and the distributivity of union and intersection also implies that the poset of subgroups of an arbitrary group ordered by containment is also a distributive lattice. You could also treat any element of the poset as an order ideal, which represents all power sets of the chosen subset.

    Might as well throw this in as well: the poset of real numbers with the standard order is also a distributive lattice, since the meet and join is given by the maximum and minimum values of two compared numbers. Take any a < b < c in the lattice. Then:

    min(a, max(b,c)) = min(a, c) = a, max(min(a,b), min(a,c)) = max(a,a) = a
    min(b, max(a,c)) = min(b,c) = b and max(min(b,a), min(b,c)) = max(a,b) = b
    min(c, max(a,b)) = min(c,b) = b and max(min(c,a), min(c,b)) = max(a,b) = b

    This also holds true if you allow for equality between any two elements.

    Each element x in this poset can map uniquely to an order ideal that contains x as the maximal element of a chain containing itself and all real numbers less than x in an infinite chain.

  3. Julia, regarding your first comment: It’s a great analogy. You are looking at “multisets” (sets with possibly repeated elements) and interpreting this lattice as being given by multiset union and intersection. To complete the proof requires a bit more care, since we’ve only discussed union and intersection for sets. You need to define what exactly “union” and “intersection” mean for multisets, and then explain why they also satisfy distributivity (as the usual union and intersection do)

    For the poset of subgroups, be careful – the join of two subgroups is not necessarily their union.

    For the reals – nice!

  4. For integers each factor should be a set (of powers of primes) I think (or is that exactly what you already said? )

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